Video: Pack 2 โ€ข Paper 2 โ€ข Question 12

Pack 2 โ€ข Paper 2 โ€ข Question 12

04:11

Video Transcript

The number of leaves on a tree at the start of the year ๐‘› is ๐ฟ ๐‘›. The number of leaves on the tree at the start of the following year can be found using the equation ๐ฟ ๐‘› plus one is equal to 0.85 multiplied by ๐ฟ ๐‘› plus five. At the start of year one, there are 105 leaves on the tree. How many leaves will there be on the tree at the start of year five? Give your answer to the nearest whole number.

Weโ€™ve been given an iterative formula which tells us how to work out the number of leaves in the year ๐‘› plus one from the number of leaves in the previous year, year ๐‘›. It tells us that we take the number of leaves at the start of the previous year ๐ฟ ๐‘›, we add five to it, and then we multiply it by 0.85. Weโ€™re told in the question that at the start of year one there are 105 leaves on the tree, which means that the value of ๐ฟ one is 105.

To find the value of ๐ฟ two, the number of leaves at the start of year two, we substitute the value of ๐ฟ one into our iterative formula, giving 0.85 multiplied by 105 plus five. We can use a calculator to evaluate this and it gives 93.5. To find the number of leaves at the start of year three, we substitute our value of ๐ฟ two into the iterative formula, giving 0.85 multiplied by 93.5 plus five. This gives 83.725.

Now, just a tip here for when you are using your calculator to evaluate these values, itโ€™s a sensible idea to use the answer button on your calculator. If you type what Iโ€™ve put on the screen here, then your calculator would take the current value add five and then multiply it by 0.85. This means that if you keep pressing equals, your calculator will keep giving you the next value in this iterative sequence.

We need to keep going because we need to work out the number of leaves in years four and year five. Iโ€™ve written down the full calculation for ๐ฟ four. But in reality, Iโ€™ve just pressed the equals button on my calculator. And it gives that ๐ฟ four is equal to 75.41625. Pressing equals one more time gives the value of ๐ฟ five 68.3538125.

The question asks us to give our answer to the nearest whole number. And as the deciding number โ€” thatโ€™s the first digit after the decimal point โ€” is a three, we round down. The number of leaves on the tree at the start of year five is 68. Now, just a note about rounding, Iโ€™ve kept the exact values on my calculator โ€” so thatโ€™s the decimals โ€” and then only rounded the answer at the end.

You may decide instead to approach this in a slightly different way. As the number of leaves on the tree each year must be an integer; that is, a whole number, you may decide to round the answers that you get for ๐ฟ two, ๐ฟ three, and so on.

The value of ๐ฟ two was 93.5. So rounding this to the nearest integer, this would give 94. Substituting 94 into the iterative formula would give the value of ๐ฟ three as 84.15. To the nearest whole number, this is just 84. Substituting 84 into the iterative formula would give the value of ๐ฟ four as 75.65, which to the nearest integer is 76. Finally, substituting 76 into the iterative formula would give the value of ๐ฟ five as 68.85. This value rounds up to 69.

This is a slightly different answer to the one we got when we only rounded at the end. But either method would be considered acceptable here. The number of leaves on the tree at the start of year five is either 68 or 69.

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